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Wavelets, Splines and Divergence-Free Vector Functions

Chapter
Part of the NATO ASI Series book series (ASIC, volume 356)

Abstract

The aim of this lecture is to give a quick review on wavelets and spline theory, a very quick one since Professor Chui already gave you an extended lecture on spline wavelets [7]. To avoid too much redundancy with Chui’s talk, I will speak as few as possible about “classical wavelet theory” - namely the orthonormal wavelet bases provided by the multiresolution analysis scheme - and a little more about heretical wavelet theories, such as bi-orthogonal wavelets or the pre-wavelets of G. Battle. A very nice example of how to apply such heretical wavelets will be given in the study of divergence-free vector wavelets.

Keywords

Multiresolution Analysis Riesz Basis Unconditional Basis Wavelet Theory Spline Wavelet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. [1]
    A. Aldroubi, M. Eden & M. Unser A family of polynomial spline wavelet transform. Preprint, 1990.Google Scholar
  2. [2]
    P. Auscher Ondelettes fractales et applications. Thèse, Paris IX, 1989.Google Scholar
  3. [3]
    G. Battle A block spin construction of ondelettes. Part I: Lemarié functions. Comm. Math. Phys. 110 (1987), 601–615.MathSciNetCrossRefGoogle Scholar
  4. [4]
    G. Battle A block spin construction of ondelettes. Part II: The QFT connection. Comm. Math. Phys. 114 (1988), 93–102.MathSciNetCrossRefGoogle Scholar
  5. [5]
    G. Battle & P. Federbush A note on divergence-free vector wavelets. Preprint, T.A.M.U., 1991.Google Scholar
  6. [6]
    G. Battle & P. Federbush Divergence-free vector wavelets. Preprint, T.A.M. U., 1991.Google Scholar
  7. [7]
    C. K. Chui An introduction to spline wavelets. Lecture at the NATO-ASI on “Approximation theory, splines and applications”, Maratea, 1991.Google Scholar
  8. [8]
    C. K. Chui & J. Z. Wang On compactly supported spline wavelets and a duality principle. To appear in Trans. Amer. Math. Soc. Google Scholar
  9. [9]
    C. K. Chui & J. Z. Wang A cardinal spline approach to wavelets. To appear in Proc. Amer. Math. Soc. Google Scholar
  10. [10]
    A. Cohen, I. Daubechies & J. C. Feauveau Bi-orthogonal bases of compactly supported wavelets. Preprint, ATT & Bell Laboratories, 1990.Google Scholar
  11. [11]
    I. Daubechies Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 46 (1988), 909–996.MathSciNetCrossRefGoogle Scholar
  12. [12]
    I. Daubechies & J. Lagarias Two-scale difference equations. Preprint, ATT & Bell Laboratories, 1989.Google Scholar
  13. [13]
    J. C. Feauveau Analyse multi-resolution par ondelettes non orthogonales et banc de filtres numériques. Thèse, Paris XI, 1990.Google Scholar
  14. [14]
    S. Jaffard Construction et propriétés des bases d’ondelettes. Thèse, Ecole Polytechnique, 1989.Google Scholar
  15. [15]
    P. G. Lemarie Construction d’ondelettes splines. Unpublished, 1987.Google Scholar
  16. [16]
    P. G. Lemarie Ondelettes à localisation exponentielle. J. Math. Pures & Appl. 67 (1988), 227–236.MathSciNetzbMATHGoogle Scholar
  17. [17]
    P. G. Lemarie Théorie L 2 des surfaces splines. Unpublished 1987.Google Scholar
  18. [18]
    P. G. Lemarie Bases d’ondelettes sur les groupes de Lie stratifiés. Bull. Soc. Math. France 117 (1989), 211–232.MathSciNetzbMATHGoogle Scholar
  19. [19]
    P. G. Lemarie Some remarks on wavelets and interpolation theory. Preprint, Paris XI, 1990.Google Scholar
  20. [20]
    P. G. Lemarie Fonctions à support compact dans les analyses multi-résolutions. To appear in Revista Matematica Ibero-americana.Google Scholar
  21. [21]
    P. G. Lemarie-Rieusset Analyses multi-résolutions non orthogonales et ondelettes vecteurs à divergence nulle. Preprint, Paris XI, 1991.Google Scholar
  22. [22]
    P. G. Lemarie & Y. Meyer Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 2 (1986), 1–18.MathSciNetCrossRefGoogle Scholar
  23. [23]
    S. Mallat A theory for multi-resolution signal decomposition: the wavelet representation. IEEE PAMI 11 (1989), 674–693.zbMATHCrossRefGoogle Scholar
  24. [24]
    Y. Meyer Ondelettes et opérateurs, tome 1. Paris, Hermann, 1990.Google Scholar
  25. [25]
    I. J. Schoenberg Cardinal spline interpolation, CBMS-NSF. Series in Applied Math. ≠ 12, SIAM Publ., Philadelphia, 1973.Google Scholar
  26. [26]
    J. O. Stromberg A modified Franklin system and higher-order systems of ℝn as unconditional bases for Hardy spaces, in Conf. on Harmonic Anal. in honor of A. Zygmund, vol. 2, Waldsworth, 1983, 475–494.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1992

Authors and Affiliations

  1. 1.CNRS UA D 0757 Université de Paris-Sud MathématiquesOrsay CedexFrance

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